Quantum coherence of thermal biphoton orbital angular momentum state and its distribution in non-Kolmogorov atmospheric turbulence

Zheng-Da Hu; Zheng-Da Hu; Zheng-Da Hu; Yun Zhu; Yun Zhu; Jicheng Wang

doi:10.1364/OE.456604

1. Introduction

Twisted photons carrying orbital angular momenta (OAM) offer a new platform for communication [1–4]. Information can be encoded with OAM beyond one bit per photon for high security [5] and capacity [6]. The potential applications of OAM in wireless communication systems have been intensively explored [7]. Direct utilizing free space as the communication channel is crucial to wireless optical communication. However, one main obstacle for free-space OAM optical communication is that the OAM modes are sensitive to the refractive index fluctuations due to atmospheric turbulence [8–10].

On the other hand, quantum entanglement is a fundamental resource in quantum communication [11]. During the last twenty years, some other quantum resources, such as quantum discord [12] and quantum coherence [13], have been found to be responsible for certain communication protocols. Quantum discord and quantum coherence can be more robust than quantum entanglement against decoherence [14]. Quantum coherence is a familiar concept usually related to interference or superposition, which lies at the root of various peculiar phenomena in quantum physics. Despite the fundamental importance of quantum coherence, a rigorous framework of quantum coherence as a source has been developed only in recent years [15]. One typical optical method for preparing quantum discord [16] is via spontaneous parametric down conversion and that for preparing quantum coherence [17] is via wave plates and beam splitters. There has already been intensive researches concerning OAM entanglement as well as its propagation via atmospheric turbulence [18–22]. Besides, distributions of orbital angular momentum multiplexed continuous-variable quantum resources in lossy or noisy channels (but without turbulence) have been experimentally reported [23,24]. By contrast, the issue of OAM quantum coherence in turbulent channels is still far from being well addressed. Very recently, a scheme to prepare thermal biphoton state with vanishing entanglement but non-zero quantum discord has been put forward [25]. With the assistance of the thermal biphoton state, quantum teleportation of a high-dimensional OAM state can be realized. To our best knowledge, the coherence property in the thermal biphoton state remains unexplored. Moreover, one needs to let one path of the thermal biphoton state be transmitted distantly to the receiver side before teleportation, in which process the effect of atmospheric turbulence may should be considered. To address the above issues, in this paper, we try to analyze the quantum coherence contained in the prepared thermal biphoton state and further consider the case of sharing quantum coherence via turbulent free space.

This work is organized as follows. In Sec. 2, the preparation of quantum thermal biphoton OAM state is introduced. In Sec. 3, the model for propagation of quantum thermal OAM states in non-Kolmogorov turbulence is presented. Section 4 gives the numerical results and discussions of turbulent effects on important quantum quantities including fidelity, quantum channel capacity and quantum coherence. Conclusions are given in Sec. 5.

2. Preparation of quantum thermal biphoton OAM state

The preparation scheme of quantum thermal biphoton OAM state is the same as that put forward by Li in Ref. [25] and here we just give a brief outline. We consider an initial biphoton state generated from chaotic thermal light source as

(1)$$\rho _{}^{(0)}=\sum_{\vec{k}_{1}}P_{\vec{k}_{1}}\vert \vec{k} _{1}\rangle \langle \vec{k}_{1}\vert \otimes \sum_{\vec{k} _{2}}P_{\vec{k}_{2}}\vert \vec{k}_{2}\rangle \langle \vec{k} _{2}\vert ,$$

where $\vert \vec {k}_{i}\rangle$ ($i=1,2$) denotes the eigenmodes of OAM for the $i$-th photon. The incoherent statistical mixture of eigenmodes $\{\vert \vec {k}_{i}\rangle \}$ for the subsystem states as well as their direct tensor product indicates that the initial state (1) is fully classical, i.e., without any quantum correlation or quantum coherence in the eigenbasis. The thermal light carrying the biphoton state $\rho ^{(0)}$ is then directed to a non-polarizing 50:50 beam splitter and divided into two paths $A$ and $B$. Assuming the reflected beam suffers a $\pi /2$ phase shift with the transformation$\vert \vec {k}_{i}\rangle \rightarrow (\vert \vec {k}_{i}\rangle _{A}+ \mathrm {i}\vert \vec {k}_{i}\rangle _{B})/\sqrt {2}$, the state $\vert \vec {k}_{1}\rangle \vert \vec {k}_{2}\rangle$ is converted to [26]

(2)$$\vert \vec{k}_{1}\rangle\vert \vec{k}_{2}\rangle \rightarrow \frac{1}{2}(\vert \vec{k}_{_{1}}\rangle_{A}\vert \vec{k}_{2}\rangle _{A}-\vert \vec{k} _{1}\rangle_{B}\vert \vec{k}_{2}\rangle_{B}+\mathrm{i}\vert \vec{k}_{1}\rangle_{A}\vert \vec{k}_{2}\rangle_{B} +\mathrm{i}\vert \vec{k}_{2}\rangle_{A}\vert \vec{k}_{1}\rangle_{B}).$$

Subsequently, an operation that postselects the last two terms to form the state $\vert \Psi (\vec {k}_{1},\vec {k}_{2})\rangle _{AB}=(\vert \vec {k}_{1}\rangle _{A}\vert \vec {k} _{2}\rangle _{B}+\vert \vec {k}_{2}\rangle _{A}\vert \vec {k}_{1}\rangle _{B})/\sqrt {2}$ is performed via the coincident measurement between photons in the two paths, which gives rise to

(3)$$\rho _{AB}^{(0)}=\sum_{\vec{k}_{1},\vec{k}_{2}}P_{\vec{k}_{1}}P_{\vec{k}_{2}}\vert \Psi (\vec{k}_{1},\vec{k}_{2})\rangle _{AB}\langle \Psi (\vec{k}_{1},\vec{k}_{2})\vert.$$

We can divide it into two parts, i.e., the diagonal terms without coherence and the non-diagonal terms containing interference (coherence) as

(4)$$\rho _{AB}^{(0)}=\sum_{\vec{k}_{1},\vec{k}_{2}}P_{\vec{k}_{1},\vec{k}_{2}}\vert \vec{k}_{1},\vec{k}_{2}\rangle_{AB}\langle \vec{k}_{1}, \vec{k}_{2}\vert +\sum_{\vec{k}_{1}\neq \vec{k}_{2}}P_{\vec{k}_{1},\vec{k}_{2}}\vert \vec{k}_{1},\vec{k}_{2}\rangle_{AB}\langle \vec{k}_{2}, \vec{k}_{1}\vert,$$

where $P_{\vec {k}_{1},\vec {k}_{2}}=P_{\vec {k}_{1}}P_{\vec {k}_{2}}$ and $\vert \vec {k}_{1},\vec {k}_{2}\rangle _{AB}=\vert \vec {k}_{1}\rangle _{A}\otimes \vert \vec {k}_{2}\rangle _{B}$. The state in Eq. (4) has been proved to be with vanishing entanglement but with nonvanishing (although little) quantum discord. The quantum discord is measured via a geometric view as $QD(\rho )=||\rho -\chi _0||^2$ [27], where $||X||^2=\mathrm {Tr}(X^\dagger X)$ denotes the Hilbert-Schmidt quadratic norm and $\chi _0$ is the classical state “nearest” to $\rho$. The analytical expression of geometric quantum discord has been derived as [25]

(5)$$QD(\rho _{AB}^{(0)})=\frac{\left(\sum_{\vec{k}}P_{\vec{k}}^{2}\right)^2-\sum_{\vec{k}}P_{\vec{k}}^{4}} {\left(\sum_{\vec{k}}P_{\vec{k}}^{2}+\left(\sum_{\vec{k}}P_{\vec{k}}\right)^{2}\right)^2},$$

where we have set the case $\vec {k}_{1},\vec {k}_{2}=\vec {k}$. Here, we will show that the quantum thermal state in Eq. (4) can have large amount of quantum coherence. To measure quantum coherence, we employ the $l_{1}$ norm of coherence, which is defined by $Coh(\rho )=\sum _{i\neq j} |\rho _{ij}|$, where $\rho _{ij}$ are the elements of state density matrix $\rho$ and $|\cdot |$ denotes the absolute value. Then, we derive the analytical expression of quantum coherence for the quantum thermal state via the $l_{1}$ norm measure as

(6)$$Coh(\rho _{AB}^{(0)})=\left(\sum_{\vec{k}}P_{\vec{k}}\right)^{2}-\sum_{\vec{k}}P_{\vec{k}}^{2},$$

If the initial state is normalized with $\sum _{\vec {k}}P_{\vec {k}}^{{}}=1$, one obtains the maximal coherence $Coh_{\max }=1-1/N$ of the quantum thermal state for $P_{\vec {k}}\equiv 1/N$, which is proportional to the dimension $N$ of Hilbert space. It is worth noting that the maximal coherence for any $N$ dimensional state is $N-1$ which can be greater than $1$ [15]. Anyhow, it can also be inferred that more coherence can be achieved when a state is encoded in higher dimension. Therefore, we will employ the OAM thermal state which theoretically offers infinite dimension. As an illustrative example, we employ the Laguerre-Gaussian (LG) beams [1], whose eigenmodes $\vert \vec {k}\rangle$ can be explicitly expressed as $\vert l,p\rangle$, where $l$ and $p$ are the OAM index and the radial index, respectively. To provide a specific expression of the probability distribution $\{P_{l,p}\}$ for the statistical mixture in the subsystem states. We further employ the Gaussian-Schell model to characterize the chaotic thermal light source owing to its generality and validity. The cross spectral density function (CSDF) of the Gaussian-Schell model can be modeled as [28]

(7)$$W^{(0)}(\vec{\rho}_{1},\vec{\rho}_{2})=\exp \left(-\frac{|\vec{\rho} _{1}|^{2}+|\vec{\rho}_{2}|^{2}}{4\sigma _{s}^{2}}-\frac{|\vec{\rho}_{1}-\vec{ \rho}_{2}|^{2}}{2\sigma _{\mu }^{2}}\right)$$

where $\sigma _{s}$ and $\sigma _{\mu }$ are positive constants determining the effective width of the spectral density and of the spectral degree of coherence across the source, respectively. By performing the mode decomposition on $W^{(0)}$ in the full set $\{\left \vert l,p\right \rangle \}$ of normalized LG modes, the probability can be found as follows [25]

(8)$$P_{l,p}=\left( 1-\tan^{4}\frac{\beta }{2}\right) \left( \tan ^{2}\frac{ \beta }{2}\right) ^{|l|+2p},$$

where $\tan \beta =2\sigma _{s}/\sigma _{\mu }$. It should be noted that the CSDF reveals spatial coherence of the beam itself while the $l_{1}$ norm of coherence measures the coherence (interference) between paths $A$ and $B$.

Figure 1 displays the $l_{1}$ norm coherence as a function of $\sigma _{\mu }/\sigma _{s}$ and the quantum discord is also displayed for comparison. We clearly observe that quantum coherence is qualitatively different from quantum discord. The quantum discord in Fig. 1(a) is always little and maximized for partially coherent beams with $\sigma _\mu /\sigma _s \neq$ $0$, $\infty$. By contrast, quantum coherence in Fig. 1(b) is a monotonic decreasing function of $\sigma _{\mu }/\sigma _{s}$ and we surprisingly find the maximum $Coh_{\mathrm {max}}=1-1/N$ is reached for the case of the completely incoherent light source with $\sigma _{\mu }=0$. From a unified view, quantum coherence is the necessary condition for the existence of quantum discord and quantum correlation can be viewed as correlated coherence [29,30]. Quantitatively comparison between them needs to be in the same framework (the same tool), although an analytical treatment for arbitrary quantum state towards this issue is still lack. Here, we have normalized the probability distribution and the dimension is specified by $N=(2L+1)(P+1)$ with $l=-L, \ldots, L$ and $p=0,\ldots, P$. For simplicity, we focus on the case of $P=0$ as usually discussed in OAM communication with LG beams.

Fig. 1. (a) Quantum discord and (b) quantum coherence as functions $\sigma _\mu /\sigma _s$ for different configurations of Hilbert space with $L=2, P=0$ (red solid), $L=2, P=1$ (red dashed), $L=10, P=0$ (blue dotted) and $L=10, P=1$ (blue dot-dashed).

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3. Propagation of quantum thermal OAM states in turbulence

After state preparation, the quantum thermal state can be shared by two persons far away from each other for quantum communication such as quantum cryptography and quantum teleportation. A convenient manner to do this is to let the one path of the biphoton connected to free space where atmospheric turbulence may exist. For simplicity, we assume the sender (Alice) prepares the quantum biphoton state, keeps photon $A$ locally and distributes photon $B$ to the receiver (Bob) via turbulent atmosphere, which is exactly the so-called one-sided noisy channel model [22]. We consider the atmospheric turbulence as non-Kolmogorov type, whose spectrum is modeled by [31]

(9)$$\begin{aligned} \Phi _{n}(\kappa ,\alpha )& =A(\alpha )\tilde{C}_{n}^{2}\frac{\exp (-\kappa ^{2}/\kappa _{m}^{2})}{(\kappa ^{2}+\kappa _{0}^{2})^{\alpha /2}},\\ 0& <\kappa <\infty ,3<\alpha <4, \end{aligned}$$

where $A(\alpha )=\Gamma (\alpha -1)\cos (\alpha \pi /2)/(4\pi ^{2})$ with $\Gamma (x)$ the Gamma function and $\alpha$ the spectral index of power law, $\tilde {C}_{n}^{2}$ is the generalized structure parameter with units $\mathrm {m}^{3-\alpha }$, $\kappa _{0}=2\pi /l_{\mathrm {out}}$ with $l_{\mathrm {out}}$ the outer scale parameter, and $\kappa _{m}=c(\alpha )/l_{\mathrm {in}}$ with $l_{\mathrm {in}}$ the inner scale parameter and $c(\alpha )=[2\pi A(\alpha )\Gamma ((5-\alpha )/2)/3]^{1/(\alpha -5)}$. For the specific case of $l_{\mathrm {out}}\rightarrow \infty$, $l_{\mathrm {in}}\rightarrow 0$ and $\alpha =11/3$, one has $A(\alpha )=0.033$ and $\tilde {C}_{n}^{2}=C_{n}^{2}$, the case of which is exactly the Kolmogorov model with $\Phi _{n}(\kappa )=0.033C_{n}^{2}\kappa ^{-11/3}$. With cylindrical coordinates $(r,\theta, z)$, the phase structure function of aberrations with spectrum (9) can be written as

(10)$$D_{\phi }(r,\delta\theta,z)=2\frac{[2r \sin (\delta\theta /2)]^{2}}{\rho _{0}^{2}},$$

where $\rho _{0}$ is the spatial coherence length of the aberrations. Under the paraxial approximation, it reads as

(11)$$\frac{1}{\rho _{0}^{2}}=\frac{1}{3}\pi ^{2}k^{2}z\int_{0}^{\infty }\kappa ^{3}\Phi _{n}(\kappa ,\alpha )\mathrm{d}\kappa =\frac{1}{3}\pi ^{2}k^{2}zT(\alpha ),$$

where

(12)$$T(\alpha )=\frac{A(\alpha )\tilde{C}_{n}^{2}}{2(\alpha -2)}\left\{\beta \kappa _{m}^{2-\alpha }\exp \left(\frac{\kappa _{0}^{2}}{\kappa _{m}^{2}}\right)\Gamma \left(2- \frac{\alpha }{2},\frac{\kappa _{0}^{2}}{\kappa _{m}^{2}}\right)-2\kappa _{0}^{4-\alpha }\right\},$$

with $\beta =2\kappa _{0}^{2}+(\alpha -2)\kappa _{m}^{2}$ and $\Gamma (\cdot,\cdot )$ the incomplete Gamma function.

Since the distribution process is characterized by the one-sided noisy channel model, the output state after the channel can be derived by applying a one-sided map $I_{A}\otimes M_{B}$ to the initially prepared state $\rho _{AB}^{(0)}$ in Eq. (4), where $I$ is the identity operator and $M=\sum _{m,n}\sqrt {P(m|n)}\left \vert m\right \rangle \left \langle n\right \vert$ is the turbulent channel operator [22] that can be constructed via the transition probabilities $P(m|n)$ from OAM eigenstates $\left \vert n\right \rangle$ to $\left \vert m\right \rangle$. The expression of $P(m|n)$ can be obtained by generalizing the results of Ref. [32] as

(13)$$P(m|n)=\frac{1}{2\pi }\int \int r\mathrm{d}r\mathrm{d}\theta \left|R(r,z)\right|^{2}\exp \left[-\left(\frac{2r \sin (\theta/2)}{\rho _{0}}\right)^2-\mathrm{i}(m-n)\theta \right],$$

where the spatial coherence length $\rho _{0}$ is determined by Eq. (11) and $R(r,z)=\left \langle r,z|l,p\right \rangle$ is the radial part of the spatial wave function of LG beam [1]. In the following simulations, we will restrict ourselves in the $2L+1$ dimensional OAM subspace but the renormalization of $P(m|n)$ after truncation is not needed.

4. Results and discussions

In this section, we begin to explore turbulence effects on important quantities in quantum information science including fidelity, quantum channel capacity (QCC) and quantum coherence. For clarity, some common parameters are set as: wavelength $\lambda =1550$ nm, beam waist $\omega _{0}=0.02$ m, generalized structure parameter $\tilde {C}_{n}^{2}=10^{-15}$ m$^{3-\alpha }$, inner scale $l_{\mathrm {in}}=10^{-3}$ m and outscale $l_{\mathrm {out}}=1$ m, unless otherwise mentioned. The beam waist should not be too small since diffraction effects are not ignorable when the Fresnel length $\sqrt {\lambda z}$ of the propagation is comparable to the beam width, although the beam with a smaller beam waist can suffer less OAM scattering [18,32]. Experimentally, the beam waist can be further magnified by a telescope [33].

To examine how close the distributed state is to the initial prepared state via the turbulent channel, Fig. (2) displays their fidelity for different spectral index $\alpha$. Here, the fidelity between two states $\rho$ and $\sigma$ can be defined by

(14)$$F=\mathrm{Tr}(\sqrt{\rho ^{1/2}\sigma \rho ^{1/2}}),$$

which satisfies several nice properties [34]. It is seen that, the fidelity exhibits monotonic attenuation as the propagation distance $z$ increases, which means the difference between the output state after turbulence and the initial state is becoming larger and larger. This is due to that the turbulence causes crosstalk of OAM modes leading to the spiral spectrum of OAM modes of the output state dramatically changed compared to that of the initial state. For simpler calculation of the fidelity, We can assume that the states before and after turbulence are $\rho$ and $\sigma$ respectively, although the fidelity is symmetric under the exchange. One can imagine that the final state at the infinite distance falls into an equal weight state as any OAM eigenstate $\left \vert l_{0}\right \rangle \rightarrow \sum _{i}$ $\left \vert l_{i}\right \rangle /\sqrt {N}$ with $N$ the dimension of Hilbert space considered. As a consequence, we can derive the far-field fidelity scaling as $F\approx 1/\sqrt {N}$. To explore the effect of the dimension of Hilbert space (total OAM modes considered), we compare the case of $L=3$ ($N=7$ modes) with $L=1$ ($N=3$ modes) and find that the fidelity will be reduced as $L$ increases. This result is consistent with that of Ref. [18]. The reason is that small OAM value corresponds to small root-mean-square beam width while the decaying probability is inversely proportional to the beam width. The smaller of the beam width, the less is the OAM crosstalk. Therefore, the mode with smaller OAM number will have higher fidelity.

Fig. 2. Fidelity as a function of propagation distance $z$ with different $\alpha$ and $L$. Here, red, blue and purple colors correspond to $\alpha =3.1$, $11/3$ and $3.9$. The solid, dashed and dotted curves correspond to $L=1$ while the circle, square and diamond are for $L=3$.

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The QCC behavior is also of interest. According to the Holevo-Schumacher-Westmoreland (HSW) theorem, the QCC can be defined by [35,36]

(15)$$QCC=\max_{\{p_{i},\rho _{i}\}}\{S(M\rho _{0}M^{{\dagger} })-\sum_{i}p_{i}S(M\rho _{i}M^{{\dagger} })\},$$

where $\rho _{0}=\sum _{i}p_{i}\rho _{i}$ is the initial state, $M$ is the noise operator, $S(\rho )=-\mathrm {Tr}(\rho \log _{2}\rho )$ is the von Neumann entropy and the maximization is run over all statistical ensembles. Here, we consider the initial state to be $\rho _{0}=\mathrm {Tr}_{A}(\rho _{AB}^{(0)})$ since only the one-sided turbulent channel $B$ is considered. The dynamics of QCC as a function of $z$ for different $\alpha$ is plotted in Fig. 3. We clearly observe that the QCC also decays with the increase of propagation distance. Nevertheless, on the contrary to the case of fidelity, the increase of $L$ can enhance the QCC because the initial QCC is $\log _{2}(2L+1)$ for $2L+1$ modes considered.

Having demonstrating the dynamics of the former two quantities (fidelity and QCC) for comparison, we would like to pay our much attention to the dynamics of quantum coherence in turbulence. In Fig. 4, we plot the evolution of coherence for different $\alpha$ and $L$. We surprisingly find that the quantum coherence exhibits qualitatively different behavior from the former two, i.e., the quantum coherence is not always decaying. By contrast, it increases at the beginning to a certain maximal value and then starts to decay. The maximal value is almost independent on $\alpha$ but its position shifts to the greater distance as the increase of $\alpha$. In addition, quantum coherence is enhanced and the maximum point is also shifted to greater distance for larger $L$. As we know, quantum coherence stems from interference (transition) between different modes. It is reasonable that larger $L$ (higher dimensional Hilbert space) can have more interference. In addition, the non-monotonic decay of coherence can be explained as an interplay between the interference effect and the decay effect of the scattering induced by the turbulence. At short distance, the decay effect is still weak and then the interference effect dominates the dynamics. In this regime, crosstalk can even contribute to the coherence. As the propagation distance increases, the decay effect turns to dominate the dynamics and then the coherence starts to decay. It is worth mentioning that distributions of OAM multiplexed quantum entanglement and quantum coherence in lossy and noisy quantum channels have been experimentally investigated [23,24], where the loss and noise are simulated by polarization beam splitters and half-wave plates. However, the effect of atmospheric turbulence on OAM is not the same as these environments. As suggested by Ref. [32], the weak atmospheric turbulence only contributes a phase abberation to the wave function (see Eq. (6) there). Besides, at the beginning of propagation, only adjacent crosstalk $\Delta m=1$ emerges (see Fig. (3) there), which ensures the dominating role is the interference not the decay effect caused by the phase aberration.

Fig. 3. Quantum channel capacity as a function of propagation distance $z$ with different $\alpha$ and $L$. Here, red, blue and purple colors are for $\alpha =3.1$, $11/3$ and $3.9$. The solid, dashed and dotted curves are for $L=1$ while the circle, square and diamond are for $L=3$.

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Fig. 4. Quantum coherence as a function of propagation distance $z$ with different $\alpha$ and $L$. Here, red, blue and purple colors are for $\alpha =3.1$, $11/3$ and $3.9$. The solid, dashed and dotted curves are for $L=1$ while the circle, square and diamond are for $L=3$.

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The case for propagating a Bell-like initial state is also presented for comparison in Fig. 5. To ensure the initial amount of coherence is the same, we consider the Bell-like initial state as $\left \vert \Psi _{0}\right \rangle = c_{+}\left \vert +L\right \rangle +c_{-}\left \vert -L\right \rangle$ with $c_{\pm }=[(3\pm \sqrt {5})/6]^{1/2}$. It is observed that the coherence of our initial superposition state is always more robust against turbulence than that of the Bell-like initial state. The maximum point is almost kept the same for the two cases. This is because our prepared state (4) is just mixture of a series Bell states of different OAM modes. More interference terms are involved in the high dimensional superposition state, which induces more coherence before the maximum point. After the maximum point, the difference between the two cases tends to shrink since the coherence of the transmitted field will be completely lost in the presence of strong turbulence at the infinite distance. The Bell-like state used for comparison is just a coherent superposition of two modes $\vert L \rangle$ and $\vert -L \rangle$ and the state after turbulence is truncated in the $2L+1$ modes subspace of $-L,-(L-1), \ldots, L-1, L$ due to crosstalk. If one considers the coherent superposition of the $2L+1$ modes, then no advantage for the quantum thermal state can be present since the quantum thermal state is incoherently mixed. However, we should remark that the preparation of such high dimensional pure state is experimentally challenging and our motivation is to illustrate the advantage of encoding information in high dimension space with the same initial resources. Besides, if one considers higher dimensional subspace after turbulence, the fidelity will be reduced due to more difference involved while the quantum coherence can be even higher due to more transition terms $P(m|n)$ added to the nondiagonal summation. The quantum channel capacity is unaffected since one may only be interested in the original $2L+1$ signal modes for the channel.

Fig. 5. Comparison of quantum coherence for the quantum thermal state (4) (solid, dashed and dotted curves) to a Bell-like pure state (circle, square and diamond) for the same initial amount of coherence. Here, red, blue and purple colors are for $\alpha =3.1$, $11/3$ and $3.9$.

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5. Conclusions

In conclusion, we have investigated the quantum coherence properties for the quantum thermal OAM state and discussed its distribution via the one-sided turbulent channel model. The channel is considered as free space with non-Kolmogorov turbulence. We find that the prepared quantum thermal OAM state has vanishing entanglement, little discord but large amount of quantum coherence. The prepared coherence is maximized when the thermal source is completely incoherent. We then consider that the one path is connected to the non-Kolmogorov turbulent atmosphere to share the quantum resource before quantum communication. The turbulence effects on important quantum quantities including fidelity, quantum channel capacity and quantum coherence are studied. We show that the behavior of quantum coherence is qualitatively different from that of the former two, the dynamics of which displays a peak during the propagation. The phenomena can be explained as the interplay between the interference effect and the decay effect of the crosstalk induced by the turbulence. In addition, we also compare the dynamics of the quantum thermal state to that of a Bell-like pure state and find that the quantum coherence can be more robust due to the fact that more interference can be induced with more OAM modes involved.

Funding

Open Foundation for CAS Key Laboratory of Quantum Information (KQI201); National Natural Science Foundation of China (11811530052, 61871202); China Science and Technology Exchange Center (CB02-20); Jiangnan University (2020366Y).

Disclosures

The authors declare no conflicts of interest.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Quantum coherence of thermal biphoton orbital angular momentum state and its distribution in non-Kolmogorov atmospheric turbulence

Abstract

1. Introduction

2. Preparation of quantum thermal biphoton OAM state

3. Propagation of quantum thermal OAM states in turbulence

4. Results and discussions

5. Conclusions

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (5)

Equations (15)

Optics Express